Maximal among abelian characteristic subgroups may be multiple and isomorphic

From Groupprops
Revision as of 20:15, 3 February 2009 by Vipul (talk | contribs) (Facts used)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Statement

It is possible to have a group of prime power order P with two distinct subgroups H, K \le P, such that both H and K are Maximal among abelian characteristic subgroups (?), and H \cong K.

Facts used

  1. Classification of abelian subgroups of maximum order in unipotent upper-triangular matrix groups

Related facts

The example described here also shows many other things:

Proof

Example involving the upper triangular matrices

Suppose p is any prime, and let G := U(5,p) be the group of upper-triangular unipotent 5 \times 5 matrices over the field of p elements. Let P be the subgroup of G comprising those matrices where the (12)^{th} entry is zero. Then, P is a group of order p^9.

By fact (1), we have that G has two subgroups that are Abelian of maximum order: the rectangle groups of dimensions 2 \times 3 and 3 \times 2 respectively. Call these subgroups H and K respectively. Then, observe that:

  • Both H and K are also Abelian subgroups of maximum order in P. Moreover, they are the only Abelian subgroups of maximum order in P since they are the only Abelian subgroups of maximum order in G.
  • H and K are isomorphic -- in fact, they are conjugate subgroups inside the bigger group GL(5,p). This conjugation restricts to an automorphism of G, but not of P.
  • Both H and K are normal in G, and hence in P. The quotient P/H is isomorphic to the unipotent subgroup of 3-by-3 matrices while the quotient P/K is isomorphic to the elementary Abelian group of order p^3. Hence, H and K are not automorphic subgroups in P.
  • Thus, H and K are the only Abelian subgroups of their order, and they are not automorphic in P. Hence, they are both characteristic subgroups. Since they are Abelian of maximum order, they are both maximal among Abelian characteristic subgroups.

We have thus established all the required conditions.